Let P(M,G) be a principal fiber bundle and E(M,N,G,P) an associated fiber bundle. Our interest is to study the harmonic sections of the projection πE of E into M. Our first purpose is give a characterization of harmonic sections of M into E regarding its equivariant lift. The second purpose is to show a version of a Liouville theorem for harmonic sections of πE.

We construct a cohomology transfer for n-fold ramified covering maps. Then we define a very general concept of transfer for ramified covering maps and prove a classification theorem for such transfers. This generalizes Roush's classification of transfers for n-fold ordinary covering maps. We characterize those representable cofunctors which admit a family of transfers for ramified covering maps that have two naturality properties, as well as normalization and stability. This is analogous to Roush's...

Sullivan associated a uniquely determined DGA|Q to any simply connected simplicial complex E. This algebra (called minimal model) contains the total (and exactly) rational homotopy information of the space E. In case E is the total space of a principal G-bundle, (G is a compact connected Lie-group) we associate a G-equivariant model UG[E], which is a collection of “G-homotopic” DGA’s|R with G-action. UG[E] will, in general, be different from the Sullivan’s minimal model of the space E. UG[E] contains the total rational...

This paper studies the relationship between the sections and the Chern or Pontrjagin classes of a vector bundle by the theory of connection. Our results are natural generalizations of the Gauss-Bonnet Theorem.